To achieve this, we have a problem that must be solved. On a filtered probability space, \((\Omega , \mathcal {F},\lbrace {\mathcal {F}}_t\rbrace _{t \ge 0},P)\) , we consider the filtrations generated by the arrival process,
\(\lbrace {\mathcal {F}}^M_t\rbrace _{t \ge 0}\) , the filtrations generated by the service process
\(\lbrace {\mathcal {F}}^S_t\rbrace _{t \ge 0}\) , and the filtrations generated by the number of the customers in the system \(\lbrace {\mathcal {F}}^N_t\rbrace _{t \ge 0}\) .
Then, we note that \({\mathcal {F}}^N_t \subseteq {\mathcal {F}}^M_t \vee {\mathcal {F}}^S_t\) , which implies that every \({\mathcal {F}}^M_t \vee {\mathcal {F}}^S_t\) -martingale is a
\({\mathcal {F}}^N_t\) -martingale.
But, the vice versa is not always true, namely, a
\({\mathcal {F}}^N_t\) -martingale is not always a \({\mathcal {F}}^M_t \vee {\mathcal {F}}^S_t\) -martingale, in general. This is the classical problem of the enlargement of filtrations.
An assumption in this framework is the
Immersion property,
which allows us to get that any
\({\mathcal {F}}^N_t\) -martingale is also a \({\mathcal {F}}^M_t \vee {\mathcal {F}}^S_t\) -martingale (see
Bielecki et al.
[1]}, Jeanblanc et al.
[2]},
and Tardelli [3]}).
Consequently, following the existing literature, the Immersion property is assumed in the present paper.
